A model structure for weakly horizontally invariant double categories

We construct a model structure on the category DblCat of double categories and double functors, whose trivial fibrations are the double functors that are surjective on objects, full on horizontal and vertical morphisms, and fully faithful on squares; and whose fibrant objects are the weakly horizontally invariant double categories. We show that the functor H^≃ :2Cat → DblCat, a more homotopical version of the usual horizontal embedding H, is right Quillen and homotopically fully faithful when considering Lack’s model structure on 2Cat. In particular, H^≃ exhibits a levelwise fibrant replacement of H. Moreover, Lack’s model structure on 2Cat is right-induced along H' from the model structure for weakly horizontally invariant double categories. We also show that this model structure is monoidal with respect to Böhm’s Gray tensor product. Finally, we prove a Whitehead theorem characterizing the weak equivalences with fibrant source as the double functors which admit a pseudoinverse up to horizontal pseudonatural equivalence.